# Flexible Body Input File

## OAFormulation

The total displacement field $\mu$ for every point of a flexible body is obtained from the displacement of a local frame defining the rigid motion of the body and from an additional local displacement field ${w}_{L}$ corresponding to the small vibrations of the body.

( ${G}_{0}$ , ${G}_{1}$ , ${G}_{2}$ , and ${G}_{3}$ ) defines the global frame ( ${e}_{1}$ , ${e}_{2}$ , and ${e}_{3}$ ).

( ${L}_{0}$ , ${L}_{1}$ , ${L}_{2}$ , and ${L}_{3}$ ) defines an orthonormal local frame.

$P$ is the rotation matrix from ( ${G}_{0}$ , ${G}_{1}$ , ${G}_{2}$ , and ${G}_{3}$ ) to ( ${L}_{0}$ , ${L}_{1}$ , ${L}_{2}$ , and ${L}_{3}$ ).

The total displacement, $u$ , can thus be expressed as:(1)
$u=X\cdot {u}_{{L}_{1}}+Y\cdot {u}_{{L}_{2}}+Z\cdot {u}_{{L}_{3}}+\left(1-X-Y-Z\right)\cdot {u}_{{L}_{0}}+P{w}_{L}={u}_{R}+P{w}_{L}$

Where, ${u}_{{L}_{0}}$ , ${u}_{{L}_{1}}$ , ${u}_{{L}_{2}}$ , and ${u}_{{L}_{3}}$ are displacements of points ${L}_{0}$ , ${L}_{1}$ , ${L}_{2}$ , and ${L}_{3}$ , respectively,

$X$ , $Y$ , and $Z$ are coordinates in the local frame ( ${L}_{0}$ , ${L}_{1}$ , ${L}_{2}$ , and ${L}_{3}$ )

${u}_{R}$ is the rigid body contribution to the total displacement

Local displacement is given by a combination of local vibration modes ${\text{Φ}}_{L}^{i}$ :(2)
${w}_{L}={\text{Φ}}_{L}\alpha$

Where, $\alpha$ is the vector of local modal contributions.

Rigid body displacement ${u}_{R}$ can also be expressed as a combination of 12 modes:(3)
${u}_{R}={\Phi }_{R}\cdot {\left({u}_{{L}_{1}}^{1},{u}_{{L}_{1}}^{2},{u}_{{L}_{1}}^{3},{u}_{{L}_{2}}^{1},{u}_{{L}_{2}}^{2},{u}_{{L}_{2}}^{3},{u}_{{L}_{3}}^{1},{u}_{{L}_{3}}^{2},{u}_{{L}_{3}}^{3},{u}_{{L}_{0}}^{1},{u}_{{L}_{0}}^{2},{u}_{{L}_{0}}^{3}\right)}^{T}$
Where the projection modes ${\Phi }_{R}^{i}$ are obtained from the local coordinates:(4)
$\begin{array}{ccc}{\mathrm{\Phi }}_{\text{R}}^{1}=\text{X}\cdot {\text{e}}_{1}& {\mathrm{\Phi }}_{\text{R}}^{2}=\text{X}\cdot {\text{e}}_{2}& {\mathrm{\Phi }}_{\text{R}}^{3}=\text{X}\cdot {\text{e}}_{3}\\ {\mathrm{\Phi }}_{\text{R}}^{4}=\text{Y}\cdot {\text{e}}_{1}& {\mathrm{\Phi }}_{\text{R}}^{5}=\text{Y}\cdot {\text{e}}_{2}& {\mathrm{\Phi }}_{\text{R}}^{6}=\text{Y}\cdot {\text{e}}_{3}\\ {\mathrm{\Phi }}_{\text{R}}^{7}=\text{Z}\cdot {\text{e}}_{1}& {\mathrm{\Phi }}_{\text{R}}^{8}=\text{Z}\cdot {\text{e}}_{2}& {\mathrm{\Phi }}_{\text{R}}^{9}=\text{Z}\cdot {\text{e}}_{3}\\ {\mathrm{\Phi }}_{\text{R}}^{10}=\left(1-\text{X}-\text{Y}-\text{Z}\right)\cdot {\text{e}}_{1}& {\mathrm{\Phi }}_{\text{R}}^{11}=\left(1-\text{X}-\text{Y}-\text{Z}\right)\cdot {\text{e}}_{2}& {\mathrm{\Phi }}_{\text{R}}^{12}=\left(1-\text{X}-\text{Y}-\text{Z}\right)\cdot {\text{e}}_{3}\end{array}$

The choice of the local frame ( ${L}_{0}$ , ${L}_{1}$ , ${L}_{2}$ , and ${L}_{3}$ ) is fully arbitrary. These points do not need to be input explicitly. Their locations define local coordinates and thus, the components of the modes ${\text{Φ}}_{R}^{i}$ .

If the flexible body contains elements with rotational DOF, three additional modes must be added to the ${\text{Φ}}_{R}^{i}$ family, accounting for the inertia associated with these DOF. The components of these additional modes on each node of the flexible body having rotational DOF are: (5)
${\mathrm{\Phi }}_{\text{R}}^{13}=\left[\begin{array}{c}0\\ 0\\ 0\\ 1\\ 0\\ 0\end{array}\right] {\mathrm{\Phi }}_{\text{R}}^{14}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 1\\ 0\end{array}\right] {\mathrm{\Phi }}_{\text{R}}^{15}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 1\end{array}\right]$